\section{Introduction}

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\frametitle{Introduction}

\begin{define}
Let $\Sigma$ be an alphabet. A picture is an two-dimensional rectangular array of elements of $\Sigma$. The set containing all images over $\Sigma$ is denoted by $\Sigma^{**}$. If L is a two dimensional language over $\Sigma$, then $L \subseteq \Sigma^{**}$. The empty image is denoted by $\lambda$. 
\end{define}

\begin{define}
Let $p \in \Sigma^{**}$ be a picture. Then 
\begin{itemize}
	\item $l_1(p)$ = number of rows of p
	\item $l_2(p)$ = number of columns of p
	\item $(l_1(p), l_2(p))$ is the size of the picture. 
	\item $p(i,j), 1 \leq i \leq l_1(p), 1 \leq j \leq l_2(p)$ or $p_{i, j}$ denotes the symbol in p with coordinates (i, j). 
	\item $\hat{p}$ is the picture p bordered by $\#'s$
	\item $B_{h, k}(p) = \{q \in \Sigma^{**} \mid l_1(q) = h, l_2(q) = k \text{ and q is subpicture of p}\}$
	\item $\hat{P}(n, m) = \{0, 1, \dots, m + 1\} \times \{0, 1, \dots, n + 1\}$
\end{itemize}
\end{define}

\end{frame}